Pat chicken thighs dry with paper towels, cut into 1-inch pieces, place in a bowl, and toss with 2 tablespoons flour until evenly coated. Season generously with salt and pepper.
Heat 2 tablespoons olive oil in Dutch oven over medium-high heat until shimmering. Working in batches, add chicken in a single layer and cook undisturbed for 3-4 minutes per side until golden brown. Transfer to a plate and set aside.
Reduce heat to medium and add remaining 1 tablespoon olive oil and 2 tablespoons butter to pot. Once butter melts, add onion, carrots, and celery with a pinch of salt and pepper. Cook for 5-6 minutes until onion is translucent, then add garlic and cook for 30 seconds until fragrant.
Sprinkle remaining 1 tablespoon flour over vegetables and stir continuously for 1-2 minutes to cook off raw flour taste and create a light roux.
Pour in chicken broth, scraping up browned bits from bottom of pot. Add halved potatoes, thyme, rosemary, sage, salt, pepper, bay leaves, and return browned chicken with accumulated juices to pot. Stir to combine.
Bring to a boil over high heat, then reduce to medium-low to maintain a gentle simmer. Partially cover and simmer for 30-35 minutes, stirring occasionally, until potatoes are fork-tender and chicken reaches 165°F internal temperature.
Remove and discard bay leaves. Taste and adjust seasoning with additional salt and pepper. If desired, stir in heavy cream and warm through for 2-3 minutes without boiling. Adjust consistency with broth if too thick or simmer uncovered if too thin.
Ladle hot stew into bowls, ensuring each serving has plenty of chicken, vegetables, and broth. Garnish with freshly chopped parsley and serve immediately with your favorite bread.